3.05/1.61	WORST_CASE(Omega(n^1), O(n^1))
3.05/1.61	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.05/1.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.05/1.61	
3.05/1.61	
3.05/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.05/1.61	
3.05/1.61	(0) CpxTRS
3.05/1.61	(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
3.05/1.61	(2) CpxTRS
3.05/1.61	    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
3.05/1.61	    (4) BOUNDS(1, n^1)
3.05/1.61	(5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.05/1.61	(6) TRS for Loop Detection
3.05/1.61	    (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.05/1.61	    (8) BEST
3.05/1.61	        (9) proven lower bound
3.05/1.61	            (10) LowerBoundPropagationProof [FINISHED, 0 ms]
3.05/1.61	            (11) BOUNDS(n^1, INF)
3.05/1.61	        (12) TRS for Loop Detection
3.05/1.61	
3.05/1.61	
3.05/1.61	----------------------------------------
3.05/1.61	
3.05/1.61	(0)
3.05/1.61	Obligation:
3.05/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.05/1.61	
3.05/1.61	
3.05/1.61	The TRS R consists of the following rules:
3.05/1.61	
3.05/1.61	   f(s(X)) -> f(X)
3.05/1.61	   g(cons(0, Y)) -> g(Y)
3.05/1.61	   g(cons(s(X), Y)) -> s(X)
3.05/1.61	   h(cons(X, Y)) -> h(g(cons(X, Y)))
3.05/1.61	
3.05/1.61	S is empty.
3.05/1.61	Rewrite Strategy: FULL
3.05/1.61	----------------------------------------
3.05/1.61	
3.05/1.61	(1) RelTrsToTrsProof (UPPER BOUND(ID))
3.05/1.61	transformed relative TRS to TRS
3.05/1.61	----------------------------------------
3.05/1.61	
3.05/1.61	(2)
3.05/1.61	Obligation:
3.05/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
3.05/1.61	
3.05/1.61	
3.05/1.61	The TRS R consists of the following rules:
3.05/1.61	
3.05/1.61	   f(s(X)) -> f(X)
3.05/1.61	   g(cons(0, Y)) -> g(Y)
3.05/1.61	   g(cons(s(X), Y)) -> s(X)
3.05/1.61	   h(cons(X, Y)) -> h(g(cons(X, Y)))
3.05/1.61	
3.05/1.61	S is empty.
3.05/1.61	Rewrite Strategy: FULL
3.05/1.61	----------------------------------------
3.05/1.61	
3.05/1.61	(3) CpxTrsMatchBoundsTAProof (FINISHED)
3.05/1.61	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 
3.05/1.61	
3.05/1.61	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
3.05/1.61	final states : [1, 2, 3]
3.05/1.61	transitions: 
3.05/1.61	s0(0) -> 0
3.05/1.61	cons0(0, 0) -> 0
3.05/1.61	00() -> 0
3.05/1.61	f0(0) -> 1
3.05/1.61	g0(0) -> 2
3.05/1.61	h0(0) -> 3
3.05/1.61	f1(0) -> 1
3.05/1.61	g1(0) -> 2
3.05/1.61	s1(0) -> 2
3.05/1.61	cons1(0, 0) -> 5
3.05/1.61	g1(5) -> 4
3.05/1.61	h1(4) -> 3
3.05/1.61	g1(0) -> 4
3.05/1.61	s1(0) -> 4
3.05/1.61	
3.05/1.61	----------------------------------------
3.05/1.61	
3.05/1.61	(4)
3.05/1.61	BOUNDS(1, n^1)
3.05/1.61	
3.05/1.61	----------------------------------------
3.05/1.61	
3.05/1.61	(5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.05/1.61	Transformed a relative TRS into a decreasing-loop problem.
3.05/1.61	----------------------------------------
3.05/1.61	
3.05/1.61	(6)
3.05/1.61	Obligation:
3.05/1.61	Analyzing the following TRS for decreasing loops:
3.05/1.61	
3.05/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.05/1.61	
3.05/1.61	
3.05/1.61	The TRS R consists of the following rules:
3.05/1.61	
3.05/1.61	   f(s(X)) -> f(X)
3.05/1.61	   g(cons(0, Y)) -> g(Y)
3.05/1.61	   g(cons(s(X), Y)) -> s(X)
3.05/1.61	   h(cons(X, Y)) -> h(g(cons(X, Y)))
3.05/1.61	
3.05/1.61	S is empty.
3.05/1.61	Rewrite Strategy: FULL
3.05/1.61	----------------------------------------
3.05/1.61	
3.05/1.61	(7) DecreasingLoopProof (LOWER BOUND(ID))
3.05/1.61	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.05/1.61	
3.05/1.61	The rewrite sequence
3.05/1.61	
3.05/1.61	g(cons(0, Y)) ->^+ g(Y)
3.05/1.61	
3.05/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.05/1.61	
3.05/1.61	The pumping substitution is [Y / cons(0, Y)].
3.05/1.61	
3.05/1.61	The result substitution is [ ].
3.05/1.61	
3.05/1.61	
3.05/1.61	
3.05/1.61	
3.05/1.61	----------------------------------------
3.05/1.61	
3.05/1.61	(8)
3.05/1.61	Complex Obligation (BEST)
3.05/1.61	
3.05/1.61	----------------------------------------
3.05/1.61	
3.05/1.61	(9)
3.05/1.61	Obligation:
3.05/1.61	Proved the lower bound n^1 for the following obligation:
3.05/1.61	
3.05/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.05/1.61	
3.05/1.61	
3.05/1.61	The TRS R consists of the following rules:
3.05/1.61	
3.05/1.61	   f(s(X)) -> f(X)
3.05/1.61	   g(cons(0, Y)) -> g(Y)
3.05/1.61	   g(cons(s(X), Y)) -> s(X)
3.05/1.61	   h(cons(X, Y)) -> h(g(cons(X, Y)))
3.05/1.61	
3.05/1.61	S is empty.
3.05/1.61	Rewrite Strategy: FULL
3.05/1.61	----------------------------------------
3.05/1.61	
3.05/1.61	(10) LowerBoundPropagationProof (FINISHED)
3.05/1.61	Propagated lower bound.
3.05/1.61	----------------------------------------
3.05/1.61	
3.05/1.61	(11)
3.05/1.61	BOUNDS(n^1, INF)
3.05/1.61	
3.05/1.61	----------------------------------------
3.05/1.61	
3.05/1.61	(12)
3.05/1.61	Obligation:
3.05/1.61	Analyzing the following TRS for decreasing loops:
3.05/1.61	
3.05/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
3.05/1.61	
3.05/1.61	
3.05/1.61	The TRS R consists of the following rules:
3.05/1.61	
3.05/1.61	   f(s(X)) -> f(X)
3.05/1.61	   g(cons(0, Y)) -> g(Y)
3.05/1.61	   g(cons(s(X), Y)) -> s(X)
3.05/1.61	   h(cons(X, Y)) -> h(g(cons(X, Y)))
3.05/1.61	
3.05/1.61	S is empty.
3.05/1.61	Rewrite Strategy: FULL
3.36/1.63	EOF